Inverse Theorems for Point-Sphere Incidences over Finite Fields
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912895895339008 |
|---|---|
| author | Singh, Shalender Singh, Vishnu Priya |
| author_facet | Singh, Shalender Singh, Vishnu Priya |
| contents | We prove the first inverse theorem for point--sphere incidence bounds over finite fields in dimensions $d \ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the structural characterization of configurations that nearly saturate these bounds has remained completely open.
Specifically, if a configuration of points $P \subset \mathbb{F}_q^d$ and spheres $\mathscr{S}$ exceeds the random incidence baseline by a factor $K$ in the moderate-sphere regime, then there exists a subset $P' \subset P$ of size \[ |P'| \gtrsim K q^{(d-1)/2} \] contained in the zero set of a polynomial $F$ of degree at most $C K^C$. This yields a one-sided result: we identify necessary algebraic obstructions to extremality, without asserting sufficiency.
The proof introduces a new rigidity mechanism for finite-field incidence geometry. Near-extremality manifests as persistent overlap among bisector hyperplanes. We prove that such persistent coincidence cannot occur without forcing the emergence of bounded-complexity algebraic certificates. The argument proceeds by isolating high-overlap layers via energy stratification, followed by a projective polynomial dichotomy applied to the set of normal directions.
As applications, we obtain the first inverse-type results for pinned distance and dot-product problems over finite fields, resolving structural questions inaccessible to standard polynomial or Fourier-analytic methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10123 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inverse Theorems for Point-Sphere Incidences over Finite Fields Singh, Shalender Singh, Vishnu Priya Combinatorics We prove the first inverse theorem for point--sphere incidence bounds over finite fields in dimensions $d \ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the structural characterization of configurations that nearly saturate these bounds has remained completely open. Specifically, if a configuration of points $P \subset \mathbb{F}_q^d$ and spheres $\mathscr{S}$ exceeds the random incidence baseline by a factor $K$ in the moderate-sphere regime, then there exists a subset $P' \subset P$ of size \[ |P'| \gtrsim K q^{(d-1)/2} \] contained in the zero set of a polynomial $F$ of degree at most $C K^C$. This yields a one-sided result: we identify necessary algebraic obstructions to extremality, without asserting sufficiency. The proof introduces a new rigidity mechanism for finite-field incidence geometry. Near-extremality manifests as persistent overlap among bisector hyperplanes. We prove that such persistent coincidence cannot occur without forcing the emergence of bounded-complexity algebraic certificates. The argument proceeds by isolating high-overlap layers via energy stratification, followed by a projective polynomial dichotomy applied to the set of normal directions. As applications, we obtain the first inverse-type results for pinned distance and dot-product problems over finite fields, resolving structural questions inaccessible to standard polynomial or Fourier-analytic methods. |
| title | Inverse Theorems for Point-Sphere Incidences over Finite Fields |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.10123 |